Socialism and the Nordic Model?

When I thought of Socialism I always thought of a welfare state. But the other day I decided to look up more about it, and I found that Socialism is actually a theory in which all means of production are publicly owned. Upon further research, discovering that in Marxist theory, it's a transitional phase into full-blown Communism. But now I'm reading that there's two types of Socialism, the one I just described, and the welfare state type, or basically the Nordic model. So is a welfare state considered a type of Socialism, or is it actually Capitalism?